不可压Navier-Stokes方程的新解耦算法

A novel fully decoupling algorithm for the incompressibleNavier-Stokes equation

非稳态不可压Navier-Stokes(NS)方程在连续意义下具有能量稳定性,在分析能量稳定性的过程中,非线性项与速度场的内积为0,这种性质被称为“零能量贡献”.利用这一性质,若引入相关的人工变量函数,在数值计算时可以显式处理非线性项且不影响能量稳定性.而不可压NS方程的传统解耦方法是引入中间变量速度场,先显式处理压力场,再通过求解类泊松方程得到原问题的速度场和压力场.将两者结合,对原方程中的非线性项和压力项均显式处理,进而得到一个对称正定的系统,因此在数值求解时可以使用共轭梯度法来提高计算效率.最后通过数值算例验证了格式的精度并和传统解耦的数值格式进行了对比.

Unsteady Navier-Stokes(NS)equations possess the energy stability in the continuous case.In the process of the energy stability analysis,the inner product of the nonlinear term and the velocity term is equal to 0.Such a property is known as'zero-energy-contribution'.Making use of this property,if a related artificial variable function is introduced,during numerical calculation the nonlinear advection term could be treated explicitly,and meanwhile its energy stability can be main-tained.Note that the traditional decoupled method is to introduce the intermediate variable velocity field,and the pressure field is first treated explicitly.Then by means of solving Poisson-type equations,the velocity field and pressure field of the original equation are obtained.By combining use of these two methods,i.e.,treating both the nonlinear terms and advection term explicitly,a symmetric positive definite system is generated.Therefore,in determining numerical solution,the conjugate gradient method can be adopted to enhance the computing efficiency.Some numerical examples are cited to verify the accuracy,and some comparisons are made in contrast with the traditional decoupled methods.